Functoriality of D-brane subcategories under generalised N=2 morphisms

Let $\Gamma$ and $\Gamma'$ be lattices with quadratic forms $q$ and $q'$, let $g:(\Gamma\oplus\Gamma^*,q)\to(\Gamma'\oplus{\Gamma'}^*,q')$ be a lattice isomorphism, and let $(\epsilon_L,\epsilon_R)=(\pm1,\pm1)$. For $(I,J)\in\mathscr{T}_{N=2}(\Gamma)$ and $\epsilon=\pm1$, write $\mathcal{C}_{I,J}^{\epsilon}(X)$ for the A- or B-brane category, with $\epsilon=-1$ for A-type and $\epsilon=1$ for B-type. Put $(I',J')=\mu_g^{(\epsilon_L,\epsilon_R)}(I,J)$. D-brane functoriality conjecture. There exists a functor

$$\Phi_{g,\epsilon}^{(\epsilon_L,\epsilon_R)}:\mathcal{C}_{I,J}^{\epsilon}(X)\longrightarrow\mathcal{C}_{I',J'}^{\epsilon_L\epsilon_R\epsilon}(X')$$

for all such $(I,J)$ and $\epsilon$, and these functors satisfy

$$\Phi_{g_2,\epsilon_{L,1}\epsilon_{R,1}\epsilon}^{(\epsilon_{L,2},\epsilon_{R,2})}\circ\Phi_{g_1,\epsilon}^{(\epsilon_{L,1},\epsilon_{R,1})}\cong\Phi_{g_2\circ g_1,\epsilon}^{(\epsilon_{L,2}\epsilon_{L,1},\epsilon_{R,2}\epsilon_{R,1})},$$

where $\cong$ denotes isomorphism of functors. This would make the A- and B-brane subcategories functorial under generalised $N=2$ morphisms, interchanging them when $\epsilon_L\epsilon_R=-1$ and preserving them when $\epsilon_L\epsilon_R=1$; the source does not report a proof.